What is a Good Metric to Study Generalization of Minimax Learners?
This work addresses a fundamental gap in understanding generalization for minimax learners, which is crucial for researchers and practitioners in machine learning dealing with adversarial training, GANs, and robust optimization, though it is incremental as it builds on existing minimax optimization frameworks.
The paper tackles the problem of identifying a suitable metric for studying generalization in stochastic minimax optimization, showing that the commonly used primal risk fails in simple cases and proposing the primal gap as a new metric. It derives generalization error bounds for this metric in nonconvex-concave settings, solves open questions regarding bounds for primal risk and primal-dual risk, and uses the metric to compare generalization behaviors of gradient descent-ascent and gradient descent-max algorithms.
Minimax optimization has served as the backbone of many machine learning (ML) problems. Although the convergence behavior of optimization algorithms has been extensively studied in the minimax settings, their generalization guarantees in stochastic minimax optimization problems, i.e., how the solution trained on empirical data performs on unseen testing data, have been relatively underexplored. A fundamental question remains elusive: What is a good metric to study generalization of minimax learners? In this paper, we aim to answer this question by first showing that primal risk, a universal metric to study generalization in minimization problems, which has also been adopted recently to study generalization in minimax ones, fails in simple examples. We thus propose a new metric to study generalization of minimax learners: the primal gap, defined as the difference between the primal risk and its minimum over all models, to circumvent the issues. Next, we derive generalization error bounds for the primal gap in nonconvex-concave settings. As byproducts of our analysis, we also solve two open questions: establishing generalization error bounds for primal risk and primal-dual risk, another existing metric that is only well-defined when the global saddle-point exists, in the strong sense, i.e., without strong concavity or assuming that the maximization and expectation can be interchanged, while either of these assumptions was needed in the literature. Finally, we leverage this new metric to compare the generalization behavior of two popular algorithms -- gradient descent-ascent (GDA) and gradient descent-max (GDMax) in stochastic minimax optimization.